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Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups

  • Yanchang Han,
    School of Mathematical Sciences, South China Normal University , Guangzhou, 510631, China
  • Yongsheng Han,
    Department of Mathematics, Auburn University , Auburn, AL 36849, USA
  • Ji Li,
    Department of Mathematics, Macquarie University , Sydney NSW 2109, Australia
  • Chaoqiang Tan,
    Department of Mathematics, Shantou University , Shantou, Guangdong 515041, China
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Abstract

The Marcinkiewicz multipliers are $L^{p}$ bounded for $1\lt p\lt \infty $ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while there is \emph{no} two parameter group of \emph{automorphic} dilations on $\mathbb{H} ^{n}$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the sense `intermediate' between the classical Lipschitz space on the Heisenberg group $\mathbb{H}^{n}$ and the product Lipschitz space on $\mathbb{C}^{n}\times \mathbb{R}$. We characterize this flag Lipschitz space via the Littlewood-Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.
Keywords: Heisenberg group, Marcinkiewicz multiplier, Flag singular integral, Flag Lipschitz space, reproducing formula, Discrete Littlewood--Paley analysis Heisenberg group, Marcinkiewicz multiplier, Flag singular integral, Flag Lipschitz space, reproducing formula, Discrete Littlewood--Paley analysis
MSC Classifications: 42B25, 42B35, 43A17 show english descriptions Maximal functions, Littlewood-Paley theory
Function spaces arising in harmonic analysis
Analysis on ordered groups, $H^p$-theory
42B25 - Maximal functions, Littlewood-Paley theory
42B35 - Function spaces arising in harmonic analysis
43A17 - Analysis on ordered groups, $H^p$-theory
 

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